Metamath Proof Explorer


Theorem qtopcmplem

Description: Lemma for qtopcmp and qtopconn . (Contributed by Mario Carneiro, 24-Mar-2015)

Ref Expression
Hypotheses qtopcmp.1 𝑋 = 𝐽
qtopcmplem.1 ( 𝐽𝐴𝐽 ∈ Top )
qtopcmplem.2 ( ( 𝐽𝐴𝐹 : 𝑋onto ( 𝐽 qTop 𝐹 ) ∧ 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )
Assertion qtopcmplem ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 qtopcmp.1 𝑋 = 𝐽
2 qtopcmplem.1 ( 𝐽𝐴𝐽 ∈ Top )
3 qtopcmplem.2 ( ( 𝐽𝐴𝐹 : 𝑋onto ( 𝐽 qTop 𝐹 ) ∧ 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )
4 simpl ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → 𝐽𝐴 )
5 simpr ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → 𝐹 Fn 𝑋 )
6 dffn4 ( 𝐹 Fn 𝑋𝐹 : 𝑋onto→ ran 𝐹 )
7 5 6 sylib ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → 𝐹 : 𝑋onto→ ran 𝐹 )
8 1 qtopuni ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋onto→ ran 𝐹 ) → ran 𝐹 = ( 𝐽 qTop 𝐹 ) )
9 2 8 sylan ( ( 𝐽𝐴𝐹 : 𝑋onto→ ran 𝐹 ) → ran 𝐹 = ( 𝐽 qTop 𝐹 ) )
10 6 9 sylan2b ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → ran 𝐹 = ( 𝐽 qTop 𝐹 ) )
11 foeq3 ( ran 𝐹 = ( 𝐽 qTop 𝐹 ) → ( 𝐹 : 𝑋onto→ ran 𝐹𝐹 : 𝑋onto ( 𝐽 qTop 𝐹 ) ) )
12 10 11 syl ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → ( 𝐹 : 𝑋onto→ ran 𝐹𝐹 : 𝑋onto ( 𝐽 qTop 𝐹 ) ) )
13 7 12 mpbid ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → 𝐹 : 𝑋onto ( 𝐽 qTop 𝐹 ) )
14 1 toptopon ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
15 2 14 sylib ( 𝐽𝐴𝐽 ∈ ( TopOn ‘ 𝑋 ) )
16 qtopid ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
17 15 16 sylan ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
18 4 13 17 3 syl3anc ( ( 𝐽𝐴𝐹 Fn 𝑋 ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )